Notation:

If $n\in \mathbb{N}$, then we denote by $\pi(n)$ the set of all non-trivial factors of $n$, including $n$ but excluding $1$. Call a set $X \subseteq \mathbb{N}$ factor-complete if it is closed under taking non-trivial factors. That is, $n\in X   \Rightarrow  \pi(n) \subseteq X$. 
Now we can state a strengthened answer to the main question of this post:

Complete characterisation:

Let $X\subseteq \mathbb{N}$. Then the following are equivalent:

$1$. $X$ is the set of orders of torsion elements of a finitely presented group $G$.

$2$. $X$ is factor-complete and has a $\Sigma_{2}^{0}$ description.

That $1\Rightarrow 2$ is immediate; torsion-orders are closed under taking factors, and have a $\Sigma_{2}^{0}$ description (see the proof of theorem 3.5 in arXiv:1107.1489v2). That $2 \Rightarrow 1$ can be seen from the following result, which is proved by a generalisation of an argument provided by Francois Dorais in an earlier post on this thread:

Technical result:

There is a uniform algorithm that, on input of a computable function $\phi: \mathbb{N}^3 \to \mathbb{N}$ which describes a factor-complete $\Sigma_{2}^{0}$ set $A$, outputs a finite presentation $P_{\phi}$ such that $A$ is precisely the set of torsion elements of $gp(P_{\phi})$.

As any set of primes is factor-complete, we get the following corollary.

Answer to main question:

A set of primes $A$ appears as the torsion elements of a finitely presented group if and only if $A$ has a $\Sigma_{2}^{0}$ description.

Observe that if $X\subseteq \mathbb{N}$ then the set $X_{prime}:=\{p_{i}\ |\ i \in X\}$ is factor-complete and one-one equivalent to $X$ (being a set of primes, with a computable numbering). So we conclude with the following:

Sets which can be realised up to one-one equivalence:
Given any $\Sigma_{2}^{0}$ set $A$, the set $A_{prime}$ is one-one equivalent to $A$, and can be realised as the set of orders of torsion elements of some finitely presented group $G$.

For completeness, we provide the proof of the technical result. This is the construction described by Francois Dorais, generalised and applied carefully so that none of the torsion elements `bump in to each other'. Apologies for its lenght.

Proof of technical result:
Let $\{a \in \mathbb{N}\ | \ (\exists m)(\forall n) (\phi(a,m,n)=1)\}$ be the description for the factor-complete $\Sigma_{2}^{0}$ set $A$, where $\phi: \mathbb{N}^3 \to \mathbb{N}$ is our computable function. 
Let $p_{1}, p_{2}, \ldots$ be the standard indexing of the primes ordered by size. 
Then we construct a countably generated recursive presentation as follows:
Take the infinite set of symbols $x_{1}, x_{2}, \ldots$; this is our generating set.
For any fixed $i>1$, add the relation $x_{p_{i}}^{i}=1$. Then, start successively computing $\phi(i,1,1), \phi(i,1,2), \phi(i,1,3), \ldots$ increasing the last variable by $1$ each time. If at some point $\phi(i,1,n)\neq 1$ then stop, add the relations $x_{p_{i}}=1$, $x_{p_{i}^{2}}^{i}=1$, and start successively computing $\phi(i,2,1), \phi(i,2,2), \phi(i,2,3), \ldots$. If again some $\phi(i,2,n)\neq 1$, then stop, add the relations $x_{p_{i}^{2}}=1$, $x_{p_{i}^{3}}^{i}=1$, and start computing $\phi(i,3,1),\phi(i,3,2), \ldots$. By interleaving this process for all $i \in \mathbb{N}$, we get a countably generated recursive presentation which we denote by $Q_{\phi}$. Notice that:

a) If $i \in A$, then there will be some (smallest) $m$ such that $\phi(i,m,1)=1, \phi(i,m,2)=1, \ldots$, and so the relation $x_{p_{i}^{m}}^{i}=1$ will be present, but no other relation involving $x_{p_{i}^{m}}$, and so $\langle x_{p_{i}^{m}} \rangle \cong C_{i}$ becomes a free product factor in this group.

b) If $i \notin A$ then for all $m \in \mathbb{N}$ we will have the relation $x_{p_{i}^{m}}=1$, and since $A$ is factor-complete $i$ does not divide any element of $A$. Hence no element of order $i$ occurs in the group.

So we end up with the group $F_{\infty}*_{a \in A}(*_{j \in \pi(a)}C_{j})$

(With a slightly more complicated indexing, one could do away with the $F_{\infty}$ factor). As $A$ is factor-complete, we immediately see that it is the set of orders of torsion elements of $gp(Q_{\phi})$. Now apply the construction of Higman-Neumann-Neumann, and then the construction of Higman, to embed this in a finitely presented group with presentation $P_{\phi}$. Note that these embeddings strictly preserve the set of orders of torsion elements (see lemma 6.9 and theorem 6.10 of M. Chiodo `Finding non-trivial elements and splittings in groups'). Also note that this construction is completely uniform in the computable function $\phi$ used to describe the set $A$.